The problem Level Planarity asks for a crossing-free drawing of a graph in the plane such that vertices are placed at prescribed y-coordinates (called levels) and such that every edge is realized as a y-monotone curve. In the variant Constrained Level Planarity, each level y is equipped with a partial order <_y on its vertices and in the desired drawing the left-to-right order of vertices on level y has to be a linear extension of <_y. Constrained Level Planarity is known to be a remarkably difficult problem: previous results by Klemz and Rote [ACM Trans. Alg. 2019] and by Br\"uckner and Rutter [SODA 2017] imply that it remains NP-hard even when restricted to graphs whose tree-depth and feedback vertex set number are bounded by a constant and even when the instances are additionally required to be either proper, meaning that each edge spans two consecutive levels, or ordered, meaning that all given partial orders are total orders. In particular, these results rule out the existence of FPT-time (even XP-time) algorithms with respect to these and related graph parameters (unless P=NP). However, the parameterized complexity of Constrained Level Planarity with respect to the vertex cover number of the input graph remained open. In this paper, we show that Constrained Level Planarity can be solved in FPT-time when parameterized by the vertex cover number. In view of the previous intractability statements, our result is best-possible in several regards: a speed-up to polynomial time or a generalization to the aforementioned smaller graph parameters is not possible, even if restricting to proper or ordered instances.

翻译：问题“层级平面性”要求在图平面中绘制无交叉的图形，使得顶点被放置在指定的y坐标（称为层级）上，并且每条边都作为y单调曲线实现。在变体“约束层级平面性”中，每个层级y上的顶点配备了一个偏序<_y，并且在所需绘图中，y层级上顶点的从左到右顺序必须是<_y的一个线性扩展。已知约束层级平面性是一个非常困难的问题：Klemz和Rote [ACM Trans. Alg. 2019] 以及Brückner和Rutter [SODA 2017] 的先前结果表明，即使当限制在树深度和反馈顶点集数有界常数，并且实例额外要求是“正常”（即每条边跨越两个连续层级）或“有序”（即所有给定的偏序都是全序）时，该问题仍然是NP难的。特别地，这些结果排除了针对这些及相关图参数存在FPT时间（甚至XP时间）算法的可能性（除非P=NP）。然而，关于输入图的顶点覆盖数的约束层级平面性的参数复杂性此前仍是开放的。在本文中，我们展示了当以顶点覆盖数为参数时，约束层级平面性可以在FPT时间内解决。鉴于先前的难解性结论，我们的结果在多个方面是最优的：即使限制为正常或有序实例，也不可能加速到多项式时间或推广到上述更小的图参数。